We are going to calculate fugacities for mixtures by using an equation of state and
in particular we will use the Peng-Robinson equation of state spreadsheet from the Elliott
and Lira textbook. And we are going to calculate fugacities and fugacity coefficients.
So given pressure, given temperature and given the mole fractions in the phase we can calculate
the fugacities. What we will do in this screencast just to calculate the fugacities and
the fugacity coefficients and show how to use the spreadsheet and then in a subsequent
screencast we will show how to calculate vapor-liquid equilibrium using the spreadsheet for
both the liquid phase and the vapor phase. In this particular example we will do the
calculation for the liquid phase. So here we are looking at the spreadsheet and first
let me point out that I made some changes. I went into review and unprotected the spreadsheet
so I could highlight what you will see in green, some of the things we are interested
in. And then I also reduced the number of significant figures just to make it a little
easier to see and understand what we are talking about. So in this column, Zi means mole fractions,
could be either liquid or vapor. We happen to be looking at a liquid. So we put in the
mole fractions of three components. And then I copied the critical values for benzene, isobutanol,
and cyclopentane. And these I could obtain or I did obtain from the Peng-Robinson equation
of state spreadsheet. Also one of the spreadsheets that is available from the Elliott and Lira
textbook website. And so I could go into that spreadsheet and copy and then paste these
values here. At 315 K and I have a pressure of 0.5 MPa, I can certainly make this lower
and still be in the liquid phase. We will look at that. And so we are interested in
liquid phase fugacities. So let me, let me highlight the liquid region and then let's
talk about that. So liquid here, one way to know this is the liquid of course is it has
a much smaller volume per mole. It is the stable phase because the fugacity of the mixture
is lower. So this fugacity in MPa and it is much lower in the liquid phase than in the
vapor phase. And then here are the three fugacity coefficients. That is just, remember, the
fugacity divided by the pressure. Here are the fugacities, at these conditions. And if
I were to lower the pressure, if I make this 0.05. Well first thing to notice, I am still
in the liquid phase because the fugacity of the mixture is lower in the liquid phase than
it is in the vapor phase. Here are the fugacities, now the fugacity coefficients change quite
a bit because they are the fugacities divided by the pressure and I changed the pressure
by a factor of 10. Perhaps didn't notice the fugacities didn't change all that much. So
let's look at fugacity for benzene, 0.0054. If I change the pressure back to 0.5, increase
it by a factor of 10, this is 0.0055 MPa, so liquid phase fugacity is very insensitive,
even if I went to 5 MPa, ah, well now we are in the one root region. So we don't talk about
liquid and vapor the same way but clearly this is a low fugacity, this is a liquid.
So we are above the critical pressure for the components but not the critical temperature.
So the fugacity changed slightly for this rather large change in pressure. And let's
go back and look at the idea that we can obtain fugacities from the equation of state. If
I were to lower the pressure enough, Let's try this. Now we are no longer in the liquid
phase region. Notice the fugacity here for the vapor phase, right the vapor phase, the
one with the larger volume, is smaller than for liquid phase, so this is the stable phase now.
And here are the fugacities for this three component mixture at this lower pressure. So you can use this
equation of state spreadsheet to determine vapor liquid equilibrium by creating a page
for liquid and another for vapor and there is a subsequent screencast that describes
how to do this.